Origami metamaterials for ultra-wideband and large-depth reflection modulation

The dynamic control of electromagnetic waves is a persistent pursuit in modern industrial development. The state-of-the-art dynamic devices suffer from limitations such as narrow bandwidth, limited modulation range, and expensive features. To address these issues, we fuse origami techniques with metamaterial design to achieve ultra-wideband and large-depth reflection modulation. Through a folding process, our proposed metamaterial achieves over 10-dB modulation depth over 4.96 – 38.8 GHz, with a fractional bandwidth of 155% and tolerance to incident angles and polarizations. Its ultra-wideband and large-depth reflection modulation performance is verified through experiments and analyzed through multipole decomposition theory. To enhance its practical applicability, transparent conductive films are introduced to the metamaterial, achieving high optical transparency (>87%) from visible to near-infrared light while maintaining cost-effectiveness. Benefiting from lightweight, foldability, and low-cost properties, our design shows promise for extensive satellite communication and optical window mobile communication management.

As illustrated in Fig. S1, an initial mechanical design with two degrees of freedom is introduced to facilitate the folding of the origami metamaterial above the solar panels.This design enables variations in the folding angle through translation while allowing for rotation, effectively rolling the origami metamaterial into the structure.In this way, the origami metamaterial is in its compressed state before the rocket is launched, reducing the storage space.Once the satellites are deployed in space, the mechanical structure effectively transforms the origami metamaterial from its compressed state to the folded state and planar state covering the whole solar panel.
The detailed deformation process can be found in Supplementary Movies 2. (3) Reconfigurable invisibility: By adjusting the folding angle of origami metamaterial, the satellite can effectively switch between visibility and invisibility state over wideband.
The expression of the wave-transfer matrix of medium and interface at arbitrary incidence angle is shown in details in Saleh's classic book 1 .With the aid of the relation between total wave-transfer matrix Mtotal and scattering matrix S, we can obtain the overall transmittance and reflectance of the multilayer system.
12 21 where the t12 and r12 are the forward amplitude transmittance and reflectance, while t21 and r21 are the amplitude transmittance and reflectance in the backward direction, respectively.In the planar state, the structure is illuminated by the light perpendicular to the sheet, while in the folded state, the normal of four parallelogram resonators of the unit cell have the same angle β with the incident wave.Hence, the transmittance and reflectance of structure at folded state can be equivalent to the transmittance and reflectance of the structure at planar state under oblique incidence, as shown in Fig. S1b.Especially, when the folded angle β = 95° of the folded structure, it can be equivalent to the light incident planar structure at an oblique angle of 52.34°.

S4: Theory of multipole decomposition
The electromagnetic waves governed by the electric field () x E and the magnetic field () x H satisfy the following time-harmonic Maxwell equations 0 0, where  and  0 are respectively permittivity and vacuum magnetic permeability.Moreover, () x J denotes the electric current density and it can be viewed as the electromagnetic source.

EE
where  E is the so-called electric far-field pattern and it is given by 3 Next, we shall use the multipole decomposition method to establish the relationship between the far-field and the electric current density.The multipole decomposition can be obtained by the long-wavelength approximation in reference 4 and we briefly describe it in the sequel.Let  denote the angle between x and y , combing equation ( 2) and the Jacobi-Anger expansion where signifies the wavenumber in vacuum, and  signifies the wavenumber in medium.Here  0 +  2 denotes the electric dipole,  1 denotes the magnetic dipole,  ̂1 +  ̂3 denotes the electric quadrupole tensor,  ̂2 denotes the magnetic quadrupole tensor,  ̂2 () denotes the electric octupole tensor and  ̂3 () is the magnetic octupole tensor.For more details, please see reference 5, 6.
We define the radiating power by Substituting ( 4) into (5)  S5b . As can be seen, the structure achieves intermittent narrowband absorption, with the absorption peak shiftings as the folded angle changes.The narrowband absorption relies on the interference cancellation between the electric ground and the lossy sheet.As a comparison, the proposed origami metamaterial, in its folded state, achieves ultra-wideband continuous absorption due to the interlayer resonance, as illustrated by multipole decomposition.

S2:
Fig. S3 | The scheme for calculating proposed structure reflectance and transmittance.a The multilayer structure corresponding to the wave-transfer matrices.b Equivalent schematic of normal incidence in folded state and oblique incidence in planar state.

Fig. S4 |
Fig. S4 | The configuration and measured results of simulated sunlight experiment.a The configuration for simulated sunlight with the solar panel placed under the illumination.b,c The scene of solar panel under the cover of proposed metamaterial at planar and folded state, respectively.The measured open circuit voltage and short circuit current are shown in the images.

By eliminating 𝐇 in ( 1
is well known that the radiating solution  to the Maxwell equations has the following asymptotic form 2

Here
are spherical Bessel functions of the first kind with order  and   are Legendre polynomials.By a direct calculation and reformulating the formula (3), one has

where
Fig. S5 | The schematic and absorption performance of the Salisbury screen with origami , thus the radiating power can be divided into different components